Det är ofta användbart att beräkna eller uppskatta kapaciteter för olika parametrar och mängder. Det kan vara till exempel vid hantering av sobolevfunktioner eller vid undersökning av partiella dierentialekvationer. För ringområden i det vanliga rummet går detta att göra exakt, men för att kunna uppskatta kapacitet i viktade rum behöver man fyra exponentmängder Q, S, Q och S till hjälp. Med dessa kan man i princip redogöra för beteendet hos kapaciteten av olika ringområden kring en x punkt. Det nns många möjliga kombinationer av hur de fyra exponentmängderna kan se ut, men det är oklart precis vilka kombinationer som är möjliga. Genom att ta fram nya exempel på kombinationer av mängderna kan vi få större kännedom om vilka kombinationer som är möjliga, och på så sätt kunna dra större nytta av dem. För att hitta nya sammansättningar utgår vi från önskade exponentmängder och undersöker om det går att ta fram bakomliggande vikter som genererar dem. Sedan tidigare nns nio exempel på kombinationer av exponentmängder. Dessa skiljer sig åt vad gäller längden på intervallen som utgör dem och om ändpunkterna tillhör intervallen eller ej. I den här rapporten har tre nya exempel på kombinationer av exponentmängder tagits fram. I alla tre exempel skiljer sig Q och S bara i ändpunkten, och vi har visat att det är möjligt att konstruera ett exempel där alla fyra mängder delar ändpunkt, men där alla mängder utom S är öppna. / It is often useful to calculate or estimate capacities for dierent parameters and sets. This is the case for example when working with Sobolev functions or when studying partial dierential equations. For annuli i Rn this can be done exactly, but when estimating capacity in weighted spaces you need four exponent sets Q, S, Q and S. With these sets it is possible to describe how the capacity of dierent annuli around a given point behaves. There are many possible combinations of the four exponent sets, and it is not clear which combinations are possible. By generating new examples of combinations of the exponent sets we obtain a larger understanding of which combinations are possible, and are thus able to use them more eciently. To nd new examples we start from the desired exponent sets and investigate if one can produce an underlying weight that generates them. Earlier, there were nine examples of combinations of exponent sets. These dier in terms of the length of the intervals that constitute them and whether the endpoints belong to the sets or not. In this thesis three new examples of combinations of exponent sets have been constructed. In all three of them, Q and S dier only in the endpoint, and we have shown that it is possible to construct an example where all four sets share the same endpoint, but all sets except S are open.

In this thesis we study radial weights on Rn. We study two radial weights with different exponent sets. We show that they are both 1-admissible by utilizing a previously shown sufficient condition, for radial weights to be 1-admissible, together with some results connecting exponent sets and Ap weights. Furthermore applying a similar method on a more general radial weight, we manage to improve the previously shown sufficient condition for radial weights to be 1-admissible. Finally we show for one of these two weights that even though it is 1-admissible, whether or not it belongs to some class Ap depends both on the value of p and on the dimension n. Additionally, both of these weights as well as another simple weight are, at least in some dimensions n, not A1 even though they are 1-admissible.

In the study of the weighted p-Laplace equation, it is often important to acquire good estimates of capacities. One useful tool for finding such estimates in metric spaces is exponent sets, which are sets describing the local dimensionality of the measure associated with the space. In this thesis, we limit ourselves to the weighted Rn space, where we investigate the relationship between exponent sets and Muckenhoupt Ap-weights - a certain class of well behaved functions. Additionally, we restrict our scope to radial weights, that is, weights w(x) that only depend on |x|. First, we determine conditions on α such that |x|α ∈ Ap(μ) for doubling measures μ on Rn. From those results, we develop weight exponent sets - a tool for making Ap-classifications of general radial weights, under certain conditions. Finally, we apply our techniques to the weight |x|α(log 1/|x|)β. We find that the weight belongs to Ap(μ) if α ∈ (-q, (p-1)q), where q = sup Q(μ) is a constant associated with the dimensionality of μ. The Ap-conditions in this thesis are found to be sharp.